Quick & dirty: plug an earphone/headset into the earphone socket. Cover the earphone/headset with a thick sweater. You're good to go.Quicker & dirtier: Maybe just plugging a bare jack would do the trick. Worth trying.

Your sound coming thru your speakers should have nothing to do with the audio captured by SnagIt. I do this often when I am recording a video and walk away, and I turn my speakers off. Have you tried recording a video that you know has audio, with your speakers off? If so, did you loose the audio You shouldn't!